Journal of Engineering Mechanics

This paper presents a testing technique to evaluate the effect of loading frequency on the behavior of structural elements. This single‐specimen technique consists in subjecting a specimen to two loading rates, but shifting, at the end of each cycle, from one frequency to the other. The writers found that by subjecting single specimens to alternate frequency loading cycles, the rate effect may be observed more clearly than by comparing the behavior of different specimens, each loaded at one of the prescribed frequencies. This is because when using a multispecimen technique, the results are prone to scatter inherent in the material properties; in particular, it would seem that when testing reinforced concrete (RC) specimens, the rate‐of‐loading effect may be of the same order of magnitude as the scatter, which is not the case when a single specimen is used. The alternate frequency technique was applied to $250\times 250\mathrm{mm}$ section RC columns subjected to uniaxial bending cycles in the range of 0.002–1 Hz. It was found that the rate effect, as manifested in the measured load and energy absorption, is about 5% higher when tested at 1 Hz than at 0.002 Hz.

The effect of prebuckling deformation on the elastic buckling of circular plates is studied. Based on Trefftz's initial stress theory, the energy functional is obtained from which the governing differential equation for equilibrium can be derived via calculus of variations. Exact and simple approximate closed‐form critical‐load expressions are obtained after solving the two‐point boundary value problem. The effect of the nonlinear terms involving in‐plane displacements in the Green‐Lagrange strain expressions on the solutions is also investigated. For circular plates with relatively large thickness‐to‐radius ratios, the allowance for prebuckling deformation affects the critical load of clamped plates more than simply supported plates. This occurs because the prebuckling deformation is greater in clamped plates, since they undergo higher loads before buckling.

The present work is concerned with the determination of the fundamental frequencies of rectangular plates carrying a mass mounted at various positions. Although direct solution of the Navier and Lévy methods is not generally practical, an energy method involving double trigonometric functions was developed to obtain approximate closed‐form solutions. A finite element analysis is also presented for comparison. Extensive experimental investigation on the three plates through a shaker system was conducted to collect useful test data. Comparisons of experimental results with theoretical predictions of the finite element and energy methods are made. The results confirm that the frequency of the plate decreases as the weight of the component mass is increased, and the frequency increases as the mass moves away from the center of the plate. The work suggested that the tested plates can be approximately represented by plates with two opposite edges simply supported and the other two edges clamped (SCSC) boundary conditions. A set of comparisons also reveals that there is a need for improvement in the analytical methods for heavier masses.

The limit loads of shallowly buried pipelines in an inelastic soil medium are investigated in this study. We model the pipelines as elastoplastic, infinitely long beams on an inelastic half space. The inelastic soil medium is characterized by the Mohr‐Coulomb and Drucker‐Prager criteria. We apply the finite‐difference method to solve the nonlinear equations of the detached pipe. The equations for the contacted pipe are solved by the finite‐element method, and the inelastic soil foundation is analyzed by the boundary‐element method with load increment. The numerical results qualitatively agree with previously published results. However, we found that the limit load of buried pipe can also be influenced by the foundation stiffness. In addition, it was found that the limit loads of buried pipes in Mohr‐Coulomb material are smaller than those in Drucker‐Prager material. The limit loads are larger for pipes with higher foundation stiffness, symmetric geometric imperfection, lower imperfection height, bigger diameter, and heavier inner content.

The bending behavior of a sandwich beam with transversely “soft” core and transverse diaphragms is analytically investigated using variational principles. The beam consists of one or two skins, upper and lower, interconnected at discrete locations through transverse diaphragms, metallic or composite laminated, with a symmetric lay‐up and a fully or partially bonded (delaminated) transverse flexible core made of foam or low‐strength honeycomb type. The diaphragm is rigidly connected to the skins but may be bonded or unbonded with the adjacent core through its entire height. The analysis considers the discrete diaphragm as a concentrated interference that affects the beam's behavior through particular boundary and continuity conditions. The behavior of the beam is determined with the aid of a high‐order theory that uses a two‐dimensional elasticity formulation for the core, in the longitudinal and the transverse directions, combined with a beam theory for the skins and the diaphragms. The core‐diaphragm interaction is determined explicitly for the bonded and the unbonded cases and the equivalent conditions in terms of the solution unknowns are derived. The presence of a discrete diaphragm yields concentrated effects that affect the beam's behavior globally and locally. These localized effects are significantly important for the safety of the structure and are explicitly determined by the analysis for the entire structure as well as in the vicinity of the diaphragm. A study that investigates the effects of the existence of a reinforcing diaphragm and its interaction with the adjacent core in a vertical cut‐off end connection between a sandwich panel and its supporting members is presented, including the level of stress concentration involved.

An exact expression for the mean outcrossing rate of a Gaussian vector process from two‐dimensional elliptical regions is derived. The expression is exact under the assumption that the cross correlation between process component 1 and velocity component 2 is zero, and it generally yields a good approximation for nonzero cross correlations. A correction for a nonzero cross correlation is shown to be vanishing to first order in the cross‐correlation coefficient. Closed‐form expressions are derived for the ensemble‐averaged rate for various types of uncertain failure surfaces, and a upper bound for the expected number of outcrossings of an exponentially deteriorating elliptical surface is determined. The analytical formulas are demonstrated to give good results by examples.

Two techniques for the artificial generation of spatially incoherent Gaussian seismic ground motions are proposed and validated. The simulated motions are homogeneous and stationary, and may be one‐, two‐, or three‐dimensional in space. They satisfy a prescribed, or target, local power spectrum and a target coherency function. Nonstationarity characteristics are introduced by superimposing a time‐dependent envelope function to produce a uniformly modulated nonstationary process. The first technique is asymptotic and approximates the coherency function by a Fourier series: it is general and suits any form of spectrum and coherency. The second technique is approximate in the sense that it satisfies the autospectrum everywhere but satisfies the cross spectrum, or the coherency, between successive stations only. The latter technique is computationally very efficient, and may be accurate enough for discretely supported systems such as single‐span structures and multispan, simply supported bridges. The techniques proposed may be useful in response analysis of structures, or structural components, for spatially incoherent random processes in the fields of earthquake, ocean, and wind engineering.

In this paper, both analytical and numerical studies of the stability of single‐degree‐of‐freedom systems with active tendon control are presented. In the studies, only an autonomous systems is considered. Prior experimental and analytical work in the area of structural control has shown that time delays are a primary cause of ineffectiveness and instability of the controlled system. Therefore, the very important effect of time delays (including on‐line computation time delay and actuator build‐up time delay) on the control force is taken into account. This is done by using a differential difference equation, or the so‐called retarded functional differential equation. Based on the concept of Liapunov stability and the method of D‐subdivision, the stable and unstable regions are determined analytically. Using numerical simulations, the accuracy of the analytical solutions are verified.

This is a theoretical study of the reflection and transmission of small‐amplitude waves by a flexible, porous, and thin beam‐like breakwater held fixed in the seabed. The fluid motion is idealized as a linearized, two‐dimensional potential flow and the breakwater is idealized as a one‐dimensional beam of uniform flexural rigidity and uniform mass per unit length. The velocity potentials of the wave motion are coupled with the equation of motion of the breakwater. Analytical solutions in closed forms are obtained for the reflected and transmitted velocity potentials together with the displacement of the breakwater. The free‐surface elevation, hydrodynamic force acting on the breakwater, and the overturning moment are determined. The dynamic response of the breakwater in terms of bending moment and shear force are also evaluated. It is found in general that hydrodynamic force increases as structural rigidity increases. The magnitude of the force is reduced dramatically for a stiffer porous breakwater. It is also shown that the stiffer structures deflect in the first mode, whereas the structures with large flexibility oscillate in higher modes. The reflection and transmission coefficients at different rigidity and porosity of a breakwater are analyzed. Results are presented to illustrate the effects of the various waves, structural parameters, and porosity parameters on the response and efficiency of the breakwater.

Assessment of fracture characteristics for cementitious composites subjected to impact loading is important in prediction of structural response and residual capacity under dynamic loading conditions. In the study reported herein, three‐point‐bend fiber‐reinforced‐concrete (FRC) specimens were tested under an impact velocity of 1.1 m/s (43 in./s). The range of specimens tested in this study included plain concrete as well as FRC reinforced with 0.5%, 1.0%, and 1.5% by volume of straight steel fibers. A new fiber‐optic sensor is introduced, by which it is possible to directly measure the displacements associated with opening of cracks in cementitious composites, if their locations are known a priori. Hence, such sensors are most suited for experimental determination of crack tip opening displacements (CTOD) in notched specimens. The development and calibration of this fiber‐optic sensor for use in cementitious composites is described. Crack tip opening displacements are measured at the tip of notched FRC specimens by embedment of fiber‐optic sensors in the specimens during mixing. The fiber‐optic sensor developed in the present study can be employed in experiments involving both static and dynamic loads.

The weak form of the equations of motion for laminated anisotropic composite shells are solved using the Ritz method for the problem of free vibration with various end conditions. Using a combination of power and Fourier series as the approximating functions for the three displacement components, the natural frequencies are evaluated for a number of geometric and material combinations. Because the full equations of elasticity are used in the formulation, no assumptions are required regarding the type of motion. The transverse shear strains, deformation of the normals, and all inertial terms are included in the formulation. The form of the approximating function assumes that the displacement components and their derivatives are continuous through the thickness of the shell. Several example geometries are considered, including a single‐layer isotropic shell, a three‐layer shell composed of dissimilar isotropic materials, a single‐layer anisotropic shell, and laminated shells composed of anisotropic materials with varying orientations. Comparisons are made with results of isotropic and anisotropic shell theories, with very good agreement being obtained.

A numerical simulation technique is presented that combines the advantages of the discrete Fourier transform (DFT) algorithm and a digital filtering scheme to generate continuous long‐duration multivariate random processes. This approach offers the simple convenience of conventional fast Fourier transform (FFT) based simulation schemes; however, it does not suffer from the drawback of the large computer memory requirement that, in the past, has precluded the generation of long‐duration time series utilizing FFT‐based approaches. Central to this technique is a simulation of a large number of time series segments by utilizing the FFT algorithm, which are subsequently synthesized by means of a digital filter to provide the desired duration of simulated processes. This approach offers computational efficiency, convenience, and robustness. The computer code based on the present methodology does not require users to have experience in determining optimal model parameters, unlike the procedures based on parametric models. The effectiveness of this methodology is demonstrated by means of examples concerning the simulation of a multivariate random wind field and the spatial variation of wave kinematics in a random sea with prescribed spectral descriptions. The simulated data showed excellent agreement with the target spectral characteristics. The proposed technique has immediate applications to the simulation of real‐time processes.

A method is proposed to compute lower bounds for the critical buckling load in problems of matrix structural analysis and in finite‐element analysis. The method is based on the fact that the critical buckling load is equal to the reciprocal of the spectral radius of a modified geometric stiffness matrix for the problem. Any matrix norm gives an upper‐bound estimate of the spectral radius, hence a lower‐bound estimate of the critical buckling load. Three easily computed types of matrix norm are used here: the maximum column‐sum norm, the maximum row‐sum norm, and the euclidean norm. By using powers of the matrix, a set of successive lower‐bound approximations can be computed that converge to the critical buckling load. A two‐bar truss, a three‐bay frame, and a finite‐element model of a plate on elastic foundation are presented as numerical examples. The last example shows how eigenvalue clustering slows convergence. The method may also be applied to estimate fundamental frequencies of vibrating structures.

The present note studies the buckling behavior of a hinged‐hinged column, taking into consideration the effect of shear deformation. The approach taken is an extension of the method proposed by Timoshenko and Gere in 1961. In this approach, the axial force is assumed to act in the direction tangent to the deformed axis of the member, and the shear force is assumed to act in the direction normal to the deformed axis. The same constitutive equation as Timoshenko and Gere is used. Further analysis differs in the following: (1) a different shear‐correction factor was used; and (2) the assumption that displacements are small was not used. As a result of latter assumption, the differential equations presented in the note are nonlinear. The solution to this equation yields not only the critical force, but the buckled shape of the rod, as well. The buckled shape is obtained by numerical integration. It is shown that by decreasing the shear stiffness (under the constant compressive force) the maximum deflection increases and, at a certain value, the possibility for higher buckling modes appears.